The technique of reconstructing a cross-sectional image of an object from multiple projections is broadly referred to as tomography. In a typical example application, a radiation source projects X-wavelength radiation through an object onto an electronic sensor array. By providing relative movement between one or more of the object, the source, and the sensor array, multiple views are obtained. An image of a slice through the object or a three-dimensional image of the object may then be approximated by use of proper mathematical transforms of the multiple views. Perhaps the best known practical application is the medical computerized tomography scanner (CT Scanner, also called computer-aided tomography or computerized axial tomography (CAT)). Tomography is also of interest in automated inspection of industrial products. However, because of cost, speed, or the mechanics required to obtain multiple images, many of the approaches to tomography that are suitable for medical applications are not suitable for a manufacturing environment. For example, consider inspection of solder joints for electronic assemblies in a production environment. There are many solder joints to be inspected, and the required test time is short. Ideally, the inspection process is in real time, as part of a feedback control system for the manufacturing process. In many manufacturing environments there is a need to verify the integrity of tens of thousands of solder joints within one minute or less.
Within X-ray absorption tomography, a number of imaging techniques are applicable to reconstruction of cross-sectional slices. In laminography, the X-ray source and sensor are moved in a coordinated fashion relative to the object to be viewed so that portions of an object outside a selected focal plane lead to a blurred image at the sensor (see, for example, U.S. Pat. No. 4,926,452). Focal plane images are reconstructed in an analog averaging process. The advantage is that extensive computer processing of ray equations is not required for image reconstruction, which makes laminography relatively fast and economical. One disadvantage is that only one plane can be reconstructed at a time (a partial solution to this problem may be found in U.S. Pat. No. 5,259,012). Another disadvantage is that out-of-focus regions, while blurred, still appear in the laminographic image, leading to shadowing and reduced contrast. Since real objects typically extend above and below the selected focal plane, laminographic smear results. Finally, constructive reinforcement between out-of-focus images can lead to artifacts in the reconstruction that may be difficult to distinguish from genuine objects.
Tomosynthesis is an approximation to laminography in which multiple projections (or views) are acquired and combined. As the number of views becomes large, the resulting combined image becomes identical to that obtained using laminography with the same geometry. A major advantage of tomosynthesis over laminography is that the focal plane to be viewed can be selected after the projections are obtained by shifting the projected images prior to recombination. Tomosynthesis may be performed as an analog method, for example, by superimposing sheets of exposed film. In digital tomosynthesis, the individual views are divided into pixels, and digitized and combined via computer software. Like laminography, however, digital tomosynthesis suffers from the drawback that all layers of an object contribute to each reconstructed cross-section leading to shadowing artifacts and reduced contrast. Partial solutions to these problems are provided by approaches in which the only areas of the images that are selected are the areas having maximum intensity or minimum intensity. Constructive artifacts still result, however, and some constructive artifacts may even be emphasized by these approaches.
Fan-beam tomography is a version of computer-aided tomography. An X-ray source having a fan-shaped planar output is positioned on one side of an object to be viewed, and a single line of sensors is positioned on the opposite side of the object to be viewed. The source and the sensors are synchronously rotated in a plane, around the stationary object. Alternatively, the object may be rotated while the source and sensor remain stationary. For objects with relatively low absorption of X-rays (such as biological tissues), this method results in a satisfactory image of a slice through the object in the plane of rotation. A three-dimensional image of the object can be obtained by electronically "stacking" a series of adjacent slices. For electronic assemblies, the plane of rotation relative to the assembly may be important. Consider, for example, a printed circuit board with lead-based solder joints in the surface plane of the board. A fan-beam slice along the plane of solder joints is subject to substantial shadowing since each ray through the board may intersect many solder joints. In addition, X-rays with sufficient energy to pass through multiple lead solder joints may damage some components in the electronic assembly. Therefore, for an electronic circuit board, it is preferable for the path of the X-rays to be at a significant angle relative to the plane of the board, preferably such that any one ray path passes through at most one lead solder joint in any one image slice. Therefore, it is preferable for the plane of rotation to be approximately orthogonal to the plane of the board. This means that the distance from the source to the sensor must be greater than the width or height of the largest object to be viewed. As discussed further below, there are other scan paths that may be more convenient for large objects in an industrial environment. It is possible to image a printed circuit board by stacking fan-beam slices that are orthogonal to the plane of the board. However, it is useful to obtain data for all the solder joints within an area at one time, for example all the solder joints for one integrated circuit. In addition, in general, the image quality from stacked slices is not as high as the image quality that can be obtained from true three-dimensional tomography.
Three-dimensional computed tomography has the potential for more accurate image reconstruction than laminography or tomosynthesis, but at the expense of speed (computation time). Three-dimensional computed tomography typically requires many projections, and is computationally intensive. One approach to three-dimensional computer-aided tomography is to position an X-ray source having a cone-shaped three-dimensional ray output on one side of an object to be viewed, and to position a two-dimensional array of sensors on the opposite side of the object to be viewed, and to synchronously move the source/array relative to the object. There are many suitable scan paths. For complete reconstruction of an arbitrary object, the scan path must surround the object. For example, the source may be moved in orthogonal circles around the object to be viewed, or the source may be moved along a helical path or other path along a cylinder surrounding the object to be viewed. This approach, called cone-beam tomography, is preferable in many cases for reconstructing three-dimensional images, and is potentially preferable for electronic assembly analysis because of the resulting image quality.
A theoretical mathematical approach to reconstructing an object from its projections was developed by J. Radon in 1917, and the basic transforms are now referred to as Radon transforms. More recently, researchers have proposed various methods for cone-beam reconstruction. See, for example:
A. K. Louis and F. Natterer, "Mathematical Problems of Computerized Tomography," Proceedings of the IEEE, Vol. 71, No.3, pp 379-389 (March 1983). PA0 R. M. Lewitt, "Reconstruction Algorithms: Transform Methods," Proceedings of the IEEE, Vol. 71, No.3, pp 390-408 (March 1983). PA0 Y. Censor, "Finite Series-Expansion Reconstruction Methods," Proceedings of the IEEE, Vol. 71, No.3, pp 409-419 (March 1983). PA0 B. D. Smith, "Cone-beam tomography: recent advances and a tutorial review," Optical Engineering, Vol. 29 No. 5, pp 524-534 (May 1990). PA0 C. Jacobson, "Fourier Methods in 3D-Reconstruction from Cone-Beam Data," Ph.D. Dissertation, Dissertation No. 427, Department of Electrical Engineering, Linkoping University, Linkoping, Sweden (1996).
In general, each method involves various trade-offs such as image quality (approximations, noise, blurring, and artifacts) versus computation time and difficulty of obtaining the required views. There is an ongoing need for economical systems with improved computation speed while providing suitable image quality.